On Siegel results about the zeros of the auxiliary function of Riemann
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913387635539968 |
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| author | de Reyna, Juan Arias |
| author_facet | de Reyna, Juan Arias |
| contents | We state and give complete proof of the results of Siegel about the zeros of the auxiliary function of Riemann $\mathop{\mathcal R}(s)$. We point out the importance of the determination of the limit to the left of the zeros of $\mathop{\mathcal R}(s)$ with positive imaginary part, obtaining the term $-\sqrt{T/2π}P(\sqrt{T/2π})$ that would explain the periodic behaviour observed with the statistical study of the zeros of $\mathop{\mathcal R}(s)$. We precise also the connection of the position on the zeros of $\mathop{\mathcal R}(s)$ with the zeros of $ζ(s)$ in the critical line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07968 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Siegel results about the zeros of the auxiliary function of Riemann de Reyna, Juan Arias Number Theory Primary 11M06, Secondary 30D99 We state and give complete proof of the results of Siegel about the zeros of the auxiliary function of Riemann $\mathop{\mathcal R}(s)$. We point out the importance of the determination of the limit to the left of the zeros of $\mathop{\mathcal R}(s)$ with positive imaginary part, obtaining the term $-\sqrt{T/2π}P(\sqrt{T/2π})$ that would explain the periodic behaviour observed with the statistical study of the zeros of $\mathop{\mathcal R}(s)$. We precise also the connection of the position on the zeros of $\mathop{\mathcal R}(s)$ with the zeros of $ζ(s)$ in the critical line. |
| title | On Siegel results about the zeros of the auxiliary function of Riemann |
| topic | Number Theory Primary 11M06, Secondary 30D99 |
| url | https://arxiv.org/abs/2406.07968 |